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The question is asking to find the formula for \( R \) in terms of \( t \), given that \( R \) is proportional to \( t^2 \).

### Step-by-step solution:
1. **Identify the relationship**: Since \( R \) is proportional to \( t^2 \), we can write:
   \[
   R = k t^2
   \]
   where \( k \) is the constant of proportionality.

2. **Use the graph to determine \( k \)**: From the graph, we can pick a point where both \( R \) and \( t \) are clearly defined. At \( t = 4 \), the graph shows that \( R = 40 \).

3. **Find the constant \( k \)**: Substituting \( R = 40 \) and \( t = 4 \) into the proportionality equation:
   \[
   40 = k (4)^2
   \]
   Simplifying:
   \[
   40 = k \cdot 16
   \]
   \[
   k = \frac{40}{16} = 2.5
   \]

4. **Final formula**: Now that we have the value of \( k \), we substitute it back into the proportionality equation:
   \[
   R = 2.5 t^2
   \]

Thus, the formula for \( R \) in terms of \( t \) is:
\[
R = 2.5 t^2
\]

Would you like more details or have any questions? Here are some related questions to expand on this topic:

1. What are other examples of quantities that are proportional to the square of another variable?
2. How does changing the constant of proportionality affect the shape of the graph?
3. Can you explain the difference between direct proportionality and inverse proportionality?
4. How would the formula change if the graph showed \( R \) proportional to \( t^3 \) instead of \( t^2 \)?
5. How can you verify that \( R = 2.5 t^2 \) holds for all points on the graph?

**Tip**: Always check units when dealing with proportionality problems in physics or real-world applications to ensure the constant makes sense in context.

Find a formula for R in terms of t, given that R is proportional to t². The graph shows the relationship between R and t for 0 ≤ t ≤ 4.

This page explains how to find the formula for R in terms of t, where R is proportional to t². A detailed step-by-step solution is provided using the graph showing the relationship between R and t for 0 ≤ t ≤ 4. Learn how to calculate the constant of proportionality (k) from the graph and derive the final formula R = 2.5t².

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